Icosahedral Polyhedra from 퐷6 lattice and Danzer’s ABCK tiling Abeer Al-Siyabi, a Nazife Ozdes Koca, a* and Mehmet Kocab aDepartment of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khoud, 123 Muscat, Sultanate of Oman, bDepartment of Physics, Cukurova University, Adana, Turkey, retired, *Correspondence e-mail: [email protected] ABSTRACT It is well known that the point group of the root lattice 퐷6 admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group 퐻3 , its roots and weights are determined in terms of those of 퐷6 . Platonic and Archimedean solids possessing icosahedral symmetry have been obtained by projections of the sets of lattice vectors of 퐷6 determined by a pair of integers (푚1, 푚2 ) in most cases, either both even or both odd. Vertices of the Danzer’s ABCK tetrahedra are determined as the fundamental weights of 퐻3 and it is shown that the inflation of the tiles can be obtained as projections of the lattice vectors characterized by the pair of integers which are linear combinations of the integers (푚1, 푚2 ) with coefficients from Fibonacci sequence. Tiling procedure both for the ABCK tetrahedral and the < 퐴퐵퐶퐾 > octahedral tilings in 3D space with icosahedral symmetry 퐻3 and those related transformations in 6D space with 퐷6 symmetry are specified by determining the rotations and translations in 3D and the corresponding group elements in 퐷6 .The tetrahedron K constitutes the fundamental region of the icosahedral group and generates the rhombic triacontahedron upon the group action. Properties of “K-polyhedron”, “B-polyhedron” and “C-polyhedron” generated by the icosahedral group have been discussed. Keywords: Lattices, Coxeter-Weyl groups, icosahedral group, projections of polytopes, polyhedra, aperiodic tilings, quasicrystals 1 1. Introduction Quasicrystallography as an emerging science attracts the interests of many scientists varying from the fields of material science, chemistry and physics. For a review see for instance the references (Di Vincenzo & Steinhardt, 1991; Janot, 1993; Senechal, 1995). It is mathematically intriguing as it requires the aperiodic tiling of the space by some prototiles. There have been several approaches to describe the aperiodicity of the quasicrystallographic space such as the set theoretic techniques, cut-and-project scheme of the higher dimensional lattices and the intuitive approaches such as the Penrose-like tilings of the space. For a review of these techniques, we propose the reference (Baake & Grimm, 2013). There have been two major approaches for the aperiodic tiling of the 3D space with local icosahedral symmetry. One of them is the Socolar-Steinhardt tiles (Socolar & Steinhardt, 1986) consisting of acute rhombohedron with golden rhombic faces, Bilinski rhombic dodecahedron, rhombic icosahedron and rhombic triacontahedron, the latter three are constructed with two Ammann tiles of acute and obtuse rhombohedra. Later it was proved that (Danzer, Papadopolos & Talis, 1993), (Roth, 1993) they can be constructed by the Danzer’s ABCK tetrahedral tiles (Danzer, 1989). Katz (Katz, 1989) and recently Hann-Socolar-Steinhardt (Hann, Socolar & Steinhardt, 2018) proposed a model of tiling scheme with decorated Ammann tiles. A detailed account of the Danzer ABCK tetrahedral tilings can be found in “math.uni- bielefeld.de/icosahedral tilings in ℝ3: the ABCK tilings” and in page 231 of the reference (Baake & Grimm, 2013) where the substitution matrix, its eigenvalues and the corresponding eigenvectors are studied. The right and left eigenvectors of the substitution matrix corresponding to the Perron-Frobenius eigenvalue are well known and will not be repeated here. Ammann rhombohedral and Danzer ABCK tetrahedral tilings are intimately related with the projection of six-dimensional cubic lattice and the root and weight lattices of 퐷6, the point symmetry group of which is of order 256!. See for a review the paper “Modelling of quasicrystals” by Kramer (Kramer, 1993) and references therein. Similar work has also been carried out in the reference (Koca, Koca & Koc, 2015). Kramer and Andrle (Kramer & Andrle, 2004) have investigated Danzer tiles from the wavelet point of view and their relations with the lattice 퐷6. In what follows we point out that the icosahedral symmetry requires a subset of the root lattice 퐷6 characterized by a pair of integers (푚1, 푚2) with 푚1 + 푚2 = 푒푣푒푛 which are the coefficients of the orthogonal set of vectors 푙, (푖 = 1, 2, … ,6). Our approach is different than the cut and project scheme of lattice 퐷6 as will be seen in the sequel. The paper consists of two major parts; first part deals with the determination of Platonic and Archimedean icosahedral polyhedra by projection of the fundamental weights of the root lattice 퐷6 into 3D space and the second part employs the technique to determine the images of the Danzer tiles in 퐷6. Inflation of the Danzer tiles are related to a redefinition of the pair of integers (푚1, 푚2) by the Fibonacci sequence. Embeddings of basic tiles in the inflated ones require translations and rotations in 3D space where the corresponding transformations in 6D space can be easily determined by the technique we have introduced. This technique which restricts the lattice 퐷6 to its subset has not been discussed elsewhere. The paper is organized as follows. In Sec. 2, we introduce the root lattice 퐷6, its icosahedral subgroup, decomposition of its weights in terms of the weights of the icosahedral group 퐻3 leading to the Archimedean polyhedra projected from the 퐷6 lattice. It turns out that the lattice vectors to be projected are determined by a pair of integers (푚1 , 푚2 ). In Sec. 3 we introduce the ABCK tetrahedral Danzer tiles in terms of the fundamental weights of the icosahedral 푛 1+√5 group 퐻 . Tiling by inflation with 휏 where 휏 = and 푛 ∈ ℤ is studied in 퐻 space by 3 2 3 prescribing the appropriate rotation and translation operators. The corresponding group elements of 퐷6 are determined noting that the pair of integers (푚1 , 푚2 ) can be expressed as the linear 2 combinations of similar integers with coefficients from Fibonacci sequence. Sec. 4 is devoted for conclusive remarks. 2. Projection of 푫ퟔ lattice under the icosahedral group 푯ퟑ and the Archimedian polyhedra We will use the Coxeter diagrams of 퐷6 and 퐻3 to introduce the basic concepts of the root systems, weights and the projection technique. A vector of the 퐷6 lattice can be written as a linear combination of the simple roots with integer coefficients: 6 6 6 휆 = ∑=1 푛훼 = ∑=1 푚푙, ∑=1 푚 = 푒푣푒푛, 푛, 푚 ∈ ℤ. (1) Here 훼 are the simple roots of 퐷6 defined in terms of the orthonormal set of vectors as 훼 = 푙 − 푙+1, 푖 = 1, … ,5 and 훼6 = 푙5 + 푙6 and the reflection generators of 퐷6 act as 푟: 푙 ⟷ 푙+1 and 푟6: 푙5 ⟷ −푙6. The generators of 퐻3 can be defined as (Koca, Koca & Koc, 2015) 푅1 = 푟1푟5, 푅2 = 푟2푟4 and 푅3 = 푟3푟6 where the Coxeter element, for example, can be taken as 푅 = 푅1푅2푅3. The weights of 퐷6 are given by ω1 = 푙1, ω2 = 푙1 + 푙2, ω3 = 푙1 + 푙2 + 푙3, ω4 = 푙1 + 푙2 + 푙3 + 푙4, 1 1 ω = (푙 + 푙 + 푙 + 푙 + 푙 − 푙 ) , ω = (푙 + 푙 + 푙 + 푙 + 푙 + 푙 ). (2) 5 2 1 2 3 4 5 6 6 2 1 2 3 4 5 6 The Voronoi cell of the root lattice 퐷6 is the dual polytope of the root polytope of ω2 (Koca et. al, 2018) and determined as the union of the orbits of the weights ω1, ω5 and ω6 which correspond to the holes of the root lattice (Conway & Sloane, 1999). If the roots and weights of ́ 퐻3 are defined in two complementary 3D spaces 퐸∥ and 퐸⊥as 훽(훽) and 푣(푣́), (푖 = 1, 2, 3) respectively then they can be expressed in terms of the roots and weights of 퐷6 as 1 1 훽 = (훼 + 휏훼 ), 푣 = (ω + 휏ω ), 1 √2+휏 1 5 1 √2+휏 1 5 1 1 훽 = (훼 + 휏훼 ), 푣 = (ω + 휏ω ), (3) 2 √2+휏 2 4 2 √2+휏 2 4 1 1 훽 = (훼 + 휏훼 ), 푣 = (ω + 휏ω ). 3 √2+휏 6 3 3 √2+휏 6 3 1+√5 For the complementary 3D space replace 훽 by 훽́ , 푣 by 푣́ in (3) and 휏 = by its algebraic 2 1− 5 conjugate 휎 = √ = −휏−1. Consequently, Cartan matrix of 퐷 (Gram matrix in lattice 2 6 terminology) and its inverse block diagonalize as 2 −1 0 0 0 0 2 −1 0 0 0 0 −1 2 −1 0 0 0 −1 2 −휏 0 0 0 0 −1 2 −1 0 0 0 −휏 2 0 0 0 퐶 = → 0 0 −1 2 −1 −1 0 0 0 2 −1 0 0 0 0 −1 2 0 0 0 0 −1 2 −휎 [ 0 0 0 −1 0 2] [ 0 0 0 0 −휎 2] (4) 3 2 2 2 2 1 1 2 + 휏 2휏2 휏3 0 0 0 2 4 4 4 2 2 2휏2 4휏2 2휏3 0 0 0 1 1 3 3 2 퐶−1 = 2 4 6 6 3 3 ⟶ 휏 2휏 3휏 0 0 0 . 2 2 4 6 8 4 4 2 0 0 0 2 + 휎 2휎2 휎3 1 2 3 4 3 2 0 0 0 2휎2 4휎2 2휎3 [1 2 3 4 2 3] [ 0 0 0 휎3 2휎3 3휎2 ] Fig.1 shows how one can illustrate the decomposition of the 6D space as the direct sum of two complementary 3D spaces where the corresponding nodes 훼, (푖 = 1, 2, . ,6), 훽, (푖 = 1, 2, 3) ́ represent the corresponding simple roots and 훽 stands for 훽 in the complementary space. Figure 1 Coxeter-Dynkin diagrams of 퐷6 and 퐻3 illustrating the symbolic projection.